Abstract
Let R be a ring with an endomorphism α and an α-derivation δ. In this article, we first compute the Jacobson radical of NI ℤ-graded rings and show that J(S) = Niℓ(S) if and only if is a ℤ-graded NI ring and J(S) ∩ S 0 is nil. As a corollary we show that, J(R[x; α]) = Niℓ(R[x; α]) if and only if R[x; α] is NI and J(R[x; α]) ∩ R ⊆ Niℓ(R). If R[x, x −1; α] is NI we prove that, J(R[x, x −1; α]) = Niℓ(R[x, x −1; α]) = Niℓ*(R[x, x −1; α]) = Niℓ(R)[x, x −1; α]. We also provide necessary and sufficient conditions for a skew polynomial ring R[x; α, δ] and skew Laurent polynomial ring R[x, x −1; α] to be NI.
ACKNOWLEDGMENT
Special thanks are due to the referee who read this article very carefully, made many useful suggestions, and recommend ℤ-graded version of Theorem 2.4, which improved the article. The author would like to thank the Banach Algebra Center of Excellence for Mathematics, University of Isfahan.
Notes
Communicated by A. Smoktunowicz.